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Stanley's theory of $(P,\omega)$-partitions is a standard tool incombinatorics. It can be extended to allow for the presence of a restriction,that is a given maximal value for partitions at each vertex of the poset, aswas shown by Assaf and Bergeron. Here we present a variation on their approach,which applies more generally. The enumerative side of the theory is morenaturally expressed in terms of back stable quasisymmetric functions. We studythe space of such functions, following the work of Lam, Lee and Shimozono onback stable symmetric functions. As applications we describe a new basis forthe ring of polynomials that we call forest polynomials. Additionally we give asigned multiplicity-free expansion for any monomial expressed in the basis ofslide polynomials.

$P$-partitions with flags and back stable quasisymmetric functions